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ic of special relativity (no gravity) in the spherically symmetric form where (23.1 ) (23.2) (1) generalized from flat spacetime Try to modify this metric to allow for curvature due to the gravitational influence of the star, while preserving spherical symmetry. The simplest and most obvious guess is to allow those met... |
rgy transport is negligible on a "hydrodynamic time scale.") Thus, it is reasonable in model building to describe the matter by perfect-fluid parameters: Material inside star to be idealized as perfect fluid p = p(r) = density of mass-energy in rest-frame of fluid; p = p(r) = isotropic pressure in rest-frame of fluid; ... |
on tells how much pressure gradient is needec! to prevent a fluid element from falling. Only the radial component of this equation has content, since the pressure depends only on r. The radial component in. the Schwarzschild coordinate system says [see the line element (23.7) and the 4~velocity components (23.13)], ---... |
uations of structure [two equations of state (23.16); equation (23.19), expressing m(r) = ~r(1 - e-211 ) as a volume integral of p; the source Equations of stellar structure summarized 606 23. SPH ERICAL STARS equation (23.21) for tP; and the OV equation of hydrostatic equilibrium (23.22)] for the five structure functi... |
hese uniform-density configurations are noteworthy. (1) For fixed energy density, Po' the central pressure (I - 2M/R)l/2 } Pc == Po 3(1 _ 2M/R)l/2 _ 1 ' 1 - { (7) increases monotonically as the radius, R, increases-and, hence, also as the mass, M == (4'17/3)PoR3, and the ratio ("strength of gravity") (8) increase. This... |
)j2 t r2 = a2 (23.34c) In the special case of a star with uniform density (Box 23.2), the entire interior is of the spherical form (23.34c); in the general case it is not. In all cases, because r> 2m(r), equation (23.34a) produces a surface with z and r as monotonically increasing functions of each other. This means th... |
, that book can be regarded as a companion volume to this one; it picks up, with astrophysical emphasis, all the topics that this book treats with gravitational emphasis. This chapter is meant only to give the reader a brief survey of the material to be found in Stars and Relativity. (continued on page 621) §24.1. OVER... |
e radiation of our sun, that is, L.=3.78X1041 ergs/sec, Calculations is indicate that the total radiation, visible and invisible, of the order L.. =101L.=3.78XI0n ergs/sec, The super nova therefore emits during its total energy E..=:::1Q6L.. =3,78XICY" If supernovae initial1y are life a ergs. r - -__ quite ordinary sta... |
p + P . 0 Here JLFe' the rest mass of an Fe56 atom, is the ratio between p + p ::::: p and n/56 in the limit of zero density. From this equation and a knowledge of p(p)-(see Figure)-one can calculate n(p). The equilibrium configurations are represented by curves of total mass-energy, M, versus radius, R. (R is defined ... |
as 1973 theory predicts, they may be subject to cracking, faulting, or slippage ("starquake") that changes the moment of inertia, and thence the rotation rate. Debris falling into the star will also change its rotation. Whichever the cause, after such a disturbance the star may rotate differentially for awhile; and how... |
y ( (24.8) (derivation in Chapter 26). The relativistic rise in the effective index of gravity above 4/3 [equation (24.6)] brings on the transition from stability (positive k; vibration) to instability (negative k; explosion or collapse) under conditions when one otherwise would have expected stability. For supermassiv... |
n here for December 19-64 (days 0.0, 1.0. 2.0, etc. in "universal time," for which see any good dictionary or encyclopedia) are taken from The American Ephemeris and Nalllical Almallac for 1964 [U.S. Government Printing Office (1962)]. y 638 25. PARTICLE MOTION IN SCHWARZSCHILD GEOMETRY Box 25.1 MASS FROM MEAN ANGULAR ... |
way was ever discovered to unite the spirit of quantum mechanics and the precision oflocation of classical mechanics. 4. Call HamiltonianH(p,x) = p2/2rn + V(x). CallenergyofparticleE. Then there _is n~~'!Y_wh~te_,,-e~consistent with the quantum principle to describe the motion of the particle in space and time. The unc... |
]; ( differenti~l scattering) cross sectlOn do dil 2'TTbdb 2'TT sin 8 de M2 (4E sin2812)2 (Rutherford). (10) (11) (12) (13) (14) (15) 648 25. PARTICLE MOTION IN SCHWARZSCHILD GEOMETRY Box 25.4 (continued) Time as correlated with position: oS o = OE = -t + fT dr [( _ M 2 £+ - - 2r 2 r 12 )]1/2· Write r = M (-2£) _ (1 - ... |
e isometry. Move each observer through the same fixed ..:1x K (e.g., leaving the manifold itself unchanged. Then have each observer report to the central computer the same geometric information as before his motion. The information received by the computer after the motion will be identical to that received before the ... |
. One of the colliding particles may be on an orbit that could never reach out to r = 00, but this makes no difference. This conservation principle allows and forces one to take over the terms E = "energy at infinity" and L = "angular momentum," valid for orbits that do reach to infinity, for an orbit that does not rea... |
aight in, or climbing Radial orbits: straight out, for then the angular momentum vanishes and the integral can be written in an elementary form that applies (with the change T --+- t) even in Newtonian mechanics, T =f dT =f dr [2M/r - 2M/R]1/2' (25.27') Here R =2M/(1 - £2) is the radius at which the particle has zero v... |
2M/RF/2 dr [2M/r - 2M/R]1I2 (1 - 2M/r) , (25.36') contains three special points: r = R, r = 0, and the added point with all the new physics, r = 2M. To admit angular momentum is to increase the number of special points still further, and to make the integral unmanageable except numerically or qualitatively (via the pot... |
ncipal features of the resulting pattern of curves. Exercise 25.21. DEFLECTION BY GRAVITY CONTRASTED WITH DEFLECTION BY ELECTRIC FORCE A test particle of arbitrary velocity 13 flies past a mass M at an impact parameter b so great that the defiection is small. Show that the defiection is 8 = ~;; (1 + 13 2 ). (25.49) Der... |
turning point; the photon comes in from infinity and ends up at r = 0: straight in for b = 0 (head-on impact); only after many loops near r = 3M, when b/M = (27)1/2 - E, where E is a very small quantity. Appreciation is expressed to Prof. R. H. Dicke for permission to publish these curves, which he had a digital calcu... |
gular momentum has the form L = ILzl = IF",I = Irp¢1 = r X ("tangential" component of 4-momentum). By symmetry, the equation L = r X ("tangential" component of p) must hold true also for orbits in other planes; it must be perfectly general: L = rp1', p1'= (tangential component of 4-momentum) = [(P0)2 + (p¢)2JlI2 (25.79... |
---- - 0.5 " f:>e'''' 't = 0.27 :o\e ~O . . . . U~",\\l. ~ o=_;;;; Collapse 0.25 0.5 0.75 1.0 --Zc~ IE bind MO + +0.02 +0.04 0.0 cluster would evolve quasistatically along a sequence ofspherical equilibrium configurations such as those of the figure. The evolution would be driven by stellar collisions and by the evapor... |
; u' (V . T) = 0; see §§22.2 and 22.3] says that dp dT = (p + p) dn dT' n Rewritten in terms of Lagrangian perturbations (recall: d/dT is a time derivative as measured by an observer moving with the fluid), this reads p + P d Lin d Lip dT -n-~' which has as its time integral (first-order analysis!) A LIp = Po + Po AO L... |
nd the critical value of the adiabatic index F 1crit is 4 Flcrit = 3" + aM/R, (9) (8) §26.6. SUMMARY OF RESULTS 699 with a a positive constant of order unity given by Expressions (8) and (9) for the pulsation frequency and the adiabatic index play an important role in the theory of supermassive stars (§24.4). 3. For al... |
state of small a, (a ~ amax) and expanding, the universe has for each a value a perfectly definite da/dt value. This velocity of expansion decreases as the expansion proceeds. It falls to zero at the turning point a = amax. Thereafter the system recontracts. §27.1. COSMOLOGY IN BRIEF 707 Lack of option is the striking... |
cluster on a small scale (clusters of galaxies of size ~3 X 10 7 light years); but one ignores the clustering. To simplify calculations, one even ignores the particulate nature of the "gas" [though one can take it into account, if one wishes, by adopting a kinetic-theory description; see §22.6 for kinetic theory, and E... |
n and receding in the opposite direction. Only an observer who is moving with the cosmo logical fluid can possibly see things as isotropic. One considers such observers in defining isotropy: Isotropy of the universe means that, at any event, an observer who is "moving with the cosmological.fluid" cannot distinguish one... |
the initial surface S[-i.e., independent of xl, x 2, x 3 • Define a(t) to be this spatially constant ratio, a(t) .10(t)/ .1o(t[). (27.18) Thus, a(t) is the factor by which the separations of world lines expand between time t[ and time t. In other words, the function a(t) is a universal "expansion factor," or "scale fa... |
pherical coordinates (R, (J, ep) and then changing to a Schwarzs child radial coordinate (2'ITr = "proper circumference"), transform this metric into the form (2727) (c) Find a further change of radial coordinate that brings the metric into the form (2722). Exercise 27.6. PROPERTIES OF THE 3-SURFACES Verify all stateme... |
and matter was §27.7. EQUATIONS OF MOTION FOR THE FLUID 727 negligible compared to Pm Vand PrV individually (see §28.l). Under these conditions, equation (2731) can be split into two parts: and The solutions are simple: (2732a) (2732b) Pm V = constant (conservation of matter) (2733a) and First law of thermodynamics use... |
by the integral of this expression from the start of the expansion: rt dt a(t); 1'/ = Jo (27.45) thus small values of the "arc parameter time," 1'/, mean early times; and larger values mean later times. In terms of this "arc-parameter measure of time," the metric takes the form (27.46) Let a photon start at the "North ... |
heat of the vacuum. At the phases of baryon-antibaryon and electron positron annihilation, the thermal gravitational radiation present has already effec tively decoupled itself from the matter, according to all current estimates. Therefore the energy set free by annihilation of matter and antimatter is expected to pou... |
is located in a limited fraction of the 3-sphere. In a short time t, according to Box 27.3, a photon can cover an arc distance on the 3-sphere equal only to 1J = (2tla*)1/2. Moreover, what is true of photons is true of other fields, forces, pressures, energies and influences: they cannot reach beyond this limit. Evide... |
ve analysis, the log-log diagram of Figure 27.5 is often more useful than the straight linear plot of V against (a/ao) of Box 27.5. All the limiting features shown in the varied types of cosmology have been encountered before in the analysis of the elementary Friedmann cosmology (big bang out of a configuration of infi... |
and the level line gives the ratio - V/ Hf,. This ratio as evaluated at any time I has the value a2(1)/a; + KJt;2, where a== da/dl. As evaluated "today" (a/ao = I) this ratio has the value I + KoH;2. Knowing the Hubble expansion rate H; today, and knowing (or trying a certain set of parameters for) the potential curve... |
which is to receive in itself all kinds [all forms] must be free from all characters [all form] .... For this reason, then, the mother and Receptacle of what has come to be visible and otherwise sensible must not be called earth or air or fire or water ... but a nature invisible and characterless, all-receiving, partak... |
he body of the sun is composed, and in receding from the sun decreases accurately as the inverse square of the distances as far as the orbit of Saturn, as evidently appears from the quiescence of the aphelion of the planets." Isaac Newton [in a letter of Dec. 10, 1692, to Richard Bentley, quoted in Munitz (1957)]: "If ... |
ive mass pulling on the sun required to produce a revolution with this period is of the order ~ 1044 g or ~lOl1M0' Age of a uranium ore as established from lead-uranium ratio: greatest value found up to 1927, 1.3 X 109 yr (A. Holmes and R. W. Lawson). Age of the lead in the "average" surface rocks of the earth as calcu... |
presently expanding and filled with matter and radiation obeying a physically acceptable equation of state must have been singular in the past, however wanting in symmetry it is today. Discovery of pulsars in 1968 by Hewish, Bell, Pilkington, Scott, and Collins, and their interpretation as spinning neutron stars (see C... |
the cosmic microwave radiation that was discovered in 1965, and they give one direct information about the nature of the universe at the time ~ 10-3, t ~ 105 years if reionization did not they last interacted with matter (a/ao occur; a/ao ~ 0.1, t ~ 5 X 108 years if reionization did occur.) Return to the history of mat... |
solution in 1973. What if the universe began chaotic? · · · . ~ :;-" r · · · · . · · The initial singularity and quantum gravitational effects Cosmologies that violate general relativity 770 28. EVOLUTION OF THE UNIVERSE INTO ITS PRESENT STATE General relativity predicts, inexorably, that even if the "initial state" w... |
ut), and must have zero rest mass. Because of the statistical nature of the process, however, some photons would lose more energy than others, and there would be a spectral broadening of the lines, which is also not observed." (3) If there does exist any such decay process, then simple arguments of special relativity t... |
when the universe was a factor of 1 + z ;:::; 1,000 smaller than it is today." [Application: In Figure 28.1, where the past evolution of the universe is sum marized, one can freely replace the horizontal scale a/ao by 1/(1 + z), and thereby see that pri mordial element formation occurred at a redshift of z ;:::; 109.) ... |
10glO z + 1.086(1 - qo)z + O(z2) - 2.5 10glO L + const. (29.34) for z ~ 1. §29.4. MAGNITUDE-REDSHIFT RELATION; MEASUREMENT OF q. 785 (Note: the factor 1.086 is actually 2.5/In 10.) A power-series solution for nonzero A (for (Jo y!:. qo) reveals a dependence on (Jo only at O(z2) and higher: R;:::; Ho-1z[ 1 - ~ (1 + qo)... |
brightest galaxy in 82 recognized regular clusters (see above). 2. The data, when fitted with a straight line, show a slope of dm/dloglOz = 5.150 + 0268 (rms), by comparison with a theoretical slope of 5. 3. The data, when fitted to the theoretical relation m = 5 10glO Z + 1.086(1 - qo)z + O(z2) + const, (5) (6) [with ... |
ace· paired with a radio telescope on earth will be able to do even better on angular resolution. Will one be able to find any fiducial distance characteristic of anyone class of objects that will serve as a natural standard of length, for very great distances (z = 2 to z = 3) as well as for galaxies closer at hand? Pe... |
ERS 799 Exercise 29.8. FRACTION OF SKY COVERED BY GALAXIES Assume that the redshifts of quasars are cosmological. Let the number of galaxies per unit physical volume in the universe today be No, and assume that no galaxies have been created or destroyed since a redshift of ?. 7. Let D be the average angular diameter of... |
oint oscillations) entail a certain minimum number of virtual quanta, which are subject to the redshifting and blueshifting action of the strong gravitational fields. Virtual quanta that are blueshifted sufficiently violently can materialize as real particles, thanks to their energy gain. In this context "sufficiently ... |
particular, shows how properties of empty space reminiscent of an elastic solid become evident near the cos mological singularity. The mathematical path to this example, as given in this box, also illustrates several important tech niques in using the variational principles for the Einstein equations to elucidate the s... |
d"A, (29) where t ="A has been written to label the specific time-coordinate choice that equation (28) implies. Mixmaster Dynamics If matter terms with no additional degrees of free dom are included, the super-Hamiltonian in equa tion (29) is modified simply. For an example, choose TOO = - TOo = (3/4)2(p.e- 3a + Fe-4a... |
me as infinite, one must attack proper time on precisely these grounds, and claim it is inadequately physical. On a local basis, where special relativity is valid, no challenge to the physical significance of proper time can succeed. It is on a more global scale that the physical primacy of proper time needs to be revi... |
efore tackling it, one must understand more fully t.han heretofore the Schwarzschild spacetime geometry, which surrounds black holes and collapsing stars as well as static stars. This chapter will concern itself with two topics that, at first sight, appear to be disconnected. One is theJaU of a test particle in. a pree... |
coordinate behavior there of the Schwarzschild metric components, gtt = - (1 - 2M/r) and grr = (1 - 2M/r)-I, must be due to a pathology there of the Schwarzschild coordinates t, r, 0, cp. Somehow one must find a way to get rid of that pathology-i.e., one must construct a new coordinate system from which the pathology ... |
to a point here, in accordance with the discussion at the end of §31.3. (r == 2M, "line" that the Kruskal-Szekeres coordinates .lthough N~yikoy's cQ_Q!"cii!).ate system i§_very _sil!!ple~2!1ceptually, ~h_e_ mathe matical -exp~~~ions fO_~Jhe metri~:~omponents in it are rath~!. ~ieldy. Simpler, more usable expressions ha... |
cosh (tI4M) v = -(rI2M - 1)1/2er/4M sinh (tI4M) u = -(1 - rI2M)1/2er/4M sinh (tI4M) v = -(1 rI2M)1/2er/4M cosh (tI4M) , , . (I) { (II) (III) (IV) { { { (31.l7a) (31.17b) (31.17c) (31.17d) Transformation between Schwarzschild coordinates and Kruskal-Szekeres coordinates *The global structure of the Schwarzschild geomet... |
anslations in the strict sense of the words only in regions II and IV, the Schwarzschild geometry. t ----+- t + Lit is a spacelike motion, not a timelike motion (see Fig. 31.3). Conse quently, a spacelike hypersurface, such as the surface t == const of Figure 31.5,a, which extends from region I through u == v == 0 into... |
ginally as the external field of a static star. One asked what happens if the star is absent; i.e., one probed the nature of the Schwarzschild geometry when no star is present to generate it. One might have expected the geometry to be that of a point mass sitting at r = O. But it was not. It turned out to represent a "... |
.1.) In place of the discarded interior Schwarzschild region, one must introduce some other coordinate system, line element, and diagram that correctly describe the interior of the collapsing star. From truncated coordinate diagrams (such as Figures 32.l,a,b,c), one can readily discover and understand the various pecul... |
(32.9b) for energy flux are different. (f) Let a collapsing star emit photons from its surface at the black-body rate dN ( - = 1.5 X 101 d'T 1 photonS) cm 2 sec K3 X (SUrfaCe area) of star X (temperature)3 of surface . Let a distant observer coUnt the photons as they pass through his sphere of radius r ~ M. Let him beg... |
ngle Xo; but it contracts from radius R = Ri = am sin XO to R = 0 as time increases. (4) The matter in the star is all crushed simultane ously to infinite density when R reaches zero, and the external Schwarzschild "funnel" de velops a cusp-like singularity at that point. ------- ~-- --~--(5) As time increases further,... |
constant ¥- 0; in the exterior give them zero rest masses, p. = 0 ("imaginary dust particles" in vacuum). Reduce the resulting metric (32.23d) to that of Friedmann inside the star, and to that of Novikov for the Schwarzschild geometry outside the star [equations (31.12)]. The effect of tidal forces on the body of a man... |
n whose circumference in EVERY direction is e :s 4'1TM (Box 32.3); (2) the external gravitational field ofa horizon (black hole), after all the "dust" and gravitational waves have cleared away, is almost certainly the Kerr-Newman generalization of the Schwarzschild geometry (Chapter 33). If so, then the external field ... |
interest implodes to form a pancake of infinite density but finite mass per unit surface area. The final kinetic energy of the dust particles is roughly equal to their final potential energy: 1 2 IV - (812'17) M M = mass of spheroid, e = circumference of final pancake. Consequently, so long as 812'17 > 2M, the collaps... |
ove result, evaluate the solution (32.31) for retarded time U > 0, using the assumptions (1) field has static form (32.32) for U < 0, ,(2) h = D' for U > o. (32.33) §32.7. NONSPHERICAL GRAVITATIONAL COLLAPSE Put the answer in the form 3 "2 MD _ D' - + --2- + 2: r r n=3 '1'1 - "2M(D' _ D)( _1)R+1(n + I)U n - 2 (2 )n r 8... |
s M, charge Q, and intrinsic angular momentum S of the collapsing star. For fully relativistic collapse, with large asym the final black hole (if one forms) is also metries and possibly a large charge, characterized uniquely by M, Q, and S. This is the conclusion that strongly suggests itself in 1972 from a set of powe... |
rection]: Relationship to Boyer-Lindquist: §33.2. PRINCIPAL FEATURES OF HOLES 879 dV = dt + (r2 + a2)(dr/fj), d;P=d</>+a(dr/fj). ds2 = - [1 - p-2(2Mr - Q2)J dV2 + 2 dr dV + p2 d82 + p-2[(r2 + a2)2 - 2ap-2(2Mr - Q2) sin28 d;P dV. fja2sin28] sin28 d;P2 - 2a sin28 d;P dr (4) (5) F = Qp-4[(r2 - a2 cos28) dr A dV - 2a2r cos... |
utside initially departs from Kerr-Newman character. Only well after the collapse occurs (asymptotic future), and in the region at and outside the hori zon, is the Kerr-Newman geometry a faithful descriptor of a black hole. This region is described in a nonsingular manner by Kerr coordinates and Kerr diagrams; and it i... |
972a); (1971), Press (1972), Hawking and Hartle (1972).] Ipser G. A star or planet falling into a large black hole will get torn apart by tidal gravita tional forces. If the tearing occurs near but outside the horizon, it may eject a blob of stellar matter that goes out with relativis tic velocity ("tube-of-toothpaste ... |
is M2 = ~r + 4M. ( Q2)2 IT S2 + 4M. 2 IT (3) [This formula, derived by Christodoulou and RUffini, may be obtained by combining equations (1), (2), and S = Ma]. E Thus, one can regard the total mass-energy of a black hole as made up of an irreducible mass, an electromagnetic mass-energy, and a rotational energy. But the... |
o the distant observer himself; i.e., relative to the hole's asymptotic Lorentz frame). (a) Show that the circuit time measured is 2'lT/ll. Thus, II can be regarded as the particle's "angular velocity as measured from infinity." (b) Let the observer moving with the particle measure its circuit time relative to the asym... |
of motion for charged test particles y EXERCISES 900 Here p2 = r2 + a2 cos2 are defined by () as defined in equation (33.3b), and the functions e, R, P 33. BLACK HOLES e = !!!. - cos2() [a2(p.2 - £2) + Lz 2/sin2()], P = £(r 2 + a2 ) - Lza - eQr, R = p2 - J[p.2r 2 + (Lz - a£)'l + !!!']. (33.33a) (33.33b) (33.33c) When w... |
Lz §33.7. STORAGE AND REMOVAL OF ENERGY 905 The infalling object will also change the direction of S. In the black hole's original asymptotic Lorentz frame, its initial angular momentum vector points in the z-direc tion, Consequently, only the z-component of angular momentum of the infalling object can produce any sig... |
root; positive square root corresponds to 4-momentum pointing toward future; while negative square root corresponds to past-pointing 4-momentum; see Figure 33.3.) Several features of the energy equation (33.54) are noteworthy. (1) For orbits in a, the energy equation yields an effective the equatorial "plane," () = 7T... |
r (l968c) proved separability for the scalar wave equation, and later when Teukolsky (1972, 1973) separated both Maxwell's equations and the wave equations for gravitational perturbations. §33.B. REVERSIBLE AND IRREVERSIBLE PROCESSES 915 Show that separation of variables for the scalar-wave equation in the (uncharged) ... |
LARITY THEOREMS J+ r = 0 singularity /+ /- r = 0 singularity /- Figure 34.3. Schwarzschild spacetime as depicted in the y. ~, n. <> coordinates of equations (34.3). This coordinate diagram should be compared with the Kruskal-Szekeres coordi nate diagram (Figure 31.3). In both diagrams, radial null geodesics are 45' lin... |
, Ne'eman (1965)]. There is no good reason to believe that the universe began with or should have begun with such strange initial structure.] Any given spacelike slice S through spacetime will intersect j-(1+) in a number of disjoint, closed, two-dimensional surfaces. Each such 2-surface is the horizon of a single blac... |
s called a "generator" of J-(!f+). 2. Follow the generator e from tJ' to ~ and then onward still further. Can it ever leave J-(!f+)? No! tJ" E ~or suppose it did leave, at some event J-(!f+). a. Repeat the entire construction of step I, with tc? conclude that there is a null the C" C J-(!f+) replacing tJ', tJ" geodesic... |
that termination point, together with all adjacent termination points, is called a "singularity." (What could be more singular than the cessation of existence for the poor tachyon, photon, or observer who moves along the terminated curve?) Another concept needed in the singularity theorems is that of a trapped surface... |
, encouraged by work of Penrose on the singularities arising in gravitational collapse, he developed new techniques which, in a series of papers in the Royal Society of London during 1966-67, established the important result that any plausible general-relativistic cosmology must be singular. The major portion of his la... |
cosmology, into gravitational collapse. Turn attention now to an idealization of an entirely different type, one independent of any symmetry considerations at all: the idealization of a "gravitational wave." Just as one identifies as "water waves" small ripples rolling across the ocean, so one gives the name "gravitati... |
35.8) defining this gauge can be summarized as (35.8d) As exercise 35.2 illustrates, only pure waves (and not more general solutions of the linearized field equations with source, Ohl'v = -16\7Tl'v) can be reduced to TT gauge. (2) for any wave Decomposition of spatial tensors 948 35. PROPAGATION OF GRAVITATIONAL WAVES ... |
o two linearly polarized components, or, alternatively, into two circularly polarized components. Displacement, /lx, for polarization w(t - z) 2m! e z • (2n + i} --- (2n + I)" (2n + t)" • ..- e. • ; • i e R i --; ..- e L ; ~ ~ .- Figure 35.1. Plane Electromagnetic Waves. Polarization vector: ep Vector Potential A = R[A... |
36.11). This makes an exactly periodic wave impossible, though a very nearly periodic one can certainly be emitted [Papapetrou (1958); Arnowitt, Deser, and Misner as reported by Misner (1964b)]. In the real universe there are spacetime curvatures due not only to the energy of gravitational waves, but also, and more im... |
transverse character of relative accelerations (2) gravitational attraction due to energy in pulse §35.11. ELECTROMAGNETIC AND GRAVITATIONAL PLANE WAVES COMPARED 961 Two particles whose separation is in the direction of propagation of the pulse (z-direction) have not only constant coordinate separation,..1x = ..1y = 0... |
expression (35.57), R~l~(h) is the only linear term, and it must vanish by itself (35.59a) [Of course hltv may contain nonlinear correction terms-call themjltv-of order d 2, which must not be constrained by this linear equation. They will be determined by (35.59c), below.] . One next splits the remainder of Rw into a ... |
ng are _(A/~)2, well below the inaccuracy of the computation. (2) Gradients average out to zero; e.g., «h lah!Lv)I/3) = O. Fractional errors made The averaging process involved in "coarse-grain" viewpoint here are ~ A/~. (3) As a corollary, one can freely integrate by parts, flipping derivatives from one h to the other... |
ion dominates, with a power output or "luminosity," L, given (see §4.4 and Figure 4.6) by Lelectric dipole = (213 )e2a2 for a single particle with acceleration a and dipole moment changing as d = eX = ea; Lelectricdipole = (2/3) d2 for a general system with dipole moment d. [Geometric units: luminosity in cm of mass-en... |
= 3.63 X 1059 ergs/sec = I! §36.3. LABORATORY GENERATORS OF GRAVITATIONAL WAVES As a laboratory generator of gravitational waves, consider a massive steel beam of radius r = 1 meter, length 1= 20 meters. density p = 7.8 g/cm3, mass M = 4.9 X 108 g (490 tons), and tensile strength t = 40,000 pounds per square inch or 3 ... |
re derived by Zerilli (1970), and were solved numerically to give these results by Davis, Ruffini, Press, and Price (1971). In the last stages of the stellar pulsations, when the amplitude ~ = Sr has dropped to Sr/r ~ 1, one can calculate the pulsation frequencies and damping times exactly by treating the fluid motions... |
ormative to extend this table to the bursts of waves emitted by (I) debris falling into a black hole, (2) collisions between two black holes, and (3) a supernova explosion in which a star of two solar masses collapses to nuclear densities, ejecting half its mass in the process. 988 36. GENERATION OF GRAVITATIONAL WAVES... |
that one cannot where ( localize the energy more closely than a wavelength!) The total power crossing a sphere of radius rat time t is (3) total power radiated (36.23) (See exercise 36.9.) This is the formula with which this chapter began: equation (36.1). The wave fronts are not precisely spherical. For example, for ... |
passing from the wave equations (36.37) to the integral equations (36.38), one has cavalierly behaved as though hJ.LV were fields in flat spacetime. This is certainly not true; but the mathematical manipulations are valid nevertheless!-and the integral equations (36.38) are valid for any field point (t, xi), even insi... |
uge conditions h/,/3 = 0; i.e., by hiO,o = h;k,k and hoo,o = hOi,i' The results are: (3) hoo and hOi in near zone calculated by gauge conditions _ [/(I)x k /(3)X k - h . = 2 _ _,_k_ + _,_k_ 2!, ,3 0, 2/(4)X k _ £ _,_k_ + 3! 3/(5)X k, ,k 4! 4/(6)X k,2] _ £ _',-,k__ 5! + (static terms not associated with radiation); h - ... |
icist. The Einsteinian physicist recognizes d 2x ijii2 as the "coordinate acceleration" of the mass element-but he keeps in mind that, to precision of O(lxiI2), coordinate lengths and proper lengths are the same [see equation (37.2)]. The coordinate acceleration d 2x ijii2 has three causes: (I) the externally applied f... |
lations in Earth-moon separation (see exercise 37.7) (c) Oscillations in Earth's crust [see Dyson (1969») (b) Normal-mode vibrations of earth and moon [see Weber (1968») (d) Normal-mode vibrations of an elastic bar [see Weber (1969) and references cited therein) (e) Normal-mode vibrations of an elas.tic square, or hoop... |
drical in shape. For a discussion of a detector with the shape of a hollow square, a hoop, or a tuning fork, see Douglass (1971); such a detector might allow its fundamental frequency to be adjdted for the most favorable response, with given mass, or given maximum dimension, or both. Sections 37.4 and 37.7 and exercise... |
tectors requires much more than the concept of cross section 1020 37. DETECTION OF GRAVITATIONAL WAVES Box 37.3 WAYS TO USE CROSS SECTION FOR WAVE·DOMINATED DETECTORS A. To Calculate Rate at which Detector Extracts Energy from a Steady Flux of Radiation ('T GW ~ 'To) I. Frequency distribution of radiation arbitrary: ( ... |
cross section is large, decreases with increasing Q. When an incoming steady flux of waves of bandwidth Llw ~ wo/Q = liTo and of specific flux optimal F.(erg/cm2 sec Hz) drives the detector, it deposits energy at the rate y 1026 ( 37. DETECTION OF GRAVITATIONAL WAVES rate of deposit) _ dE - f of energy dt _ _ resonance... |
nflicting notations. Considering the way in which the best notation that is available today for spin-l radiation was evolved, one can only feel that it is too early to canonize anyone notation for describing the scattering parameters for an object that is scattering gravitational radiation. Exercise 37.8. POWER RERADIA... |
(c) Show that the reduced quadrupole moments are (37.46c) (37.46d) (37.46e) (d) Show that the rate of emission of vibrational energy, averaged over a period, is (37.46f) (e) Show that the exponential rate of decay of energy by reason of gravitational wave damping, or "gravitational radiation line broadening," is (37.46... |
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